Some properties of Stepanov-like almost automorphic functions and applications to abstract evolution equations

Abstract

This article is concerned with some properties of Stepanov-like almost automorphic (S ^p -a.a.) functions. We establish a composition theorem about S ^p -a.a. functions, and with its help, study the existence and uniqueness of almost automorphic solutions for semilinear evolution equations in Banach spaces. Moreover, integration and differentiation of S ^p -a.a. functions are discussed. Some theorems extend earlier results.

Keywords:

• almost automorphic
• Stepanov-like almost automorphic
• composition theorem
• evolution equations
• Banach space

AMS Subject Classifications:

• 43A60
• 34G20

Acknowledgements

Hui-Sheng Ding acknowledges support from the NSF of China, the NSF of Jiangxi Province of China (2008GQS0057) and the Youth Foundation of Jiangxi Provincial Education Department (GJJ09456). Jin Liang and Ti-Jun Xiao acknowledge support from the NSF of China (10771202), the Research Fund for Shanghai Key Laboratory of Modern Applied Mathematics (08DZ2271900) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (2007035805).